Problem 101 - Precalculus Essentials

Question

Use a graphing utility to graph each equation. Then use the \([\text { TRACE }]\) feature to trace along the line and find the coordinates of two points. Use these points to compute the line's slope. Check your result by using the coefficient of \(x\) in the line's equation. $$y=-3 x+6$$

Step-by-Step Solution

Answer

The calculated slope of the line \(y=-3x+6\) is -3 which matches with the coefficient of \(x\) in the equation. The line's slope is therefore -3.

Step 1: Graph the Equation

A graphing utility will be employed to plot the line using the equation \(y=-3x+6\). This will give us a visual representation of the line to allow us to pick two points.

Step 2: Trace and Pick Two Points

The TRACE feature will be used on the graphing utility to move along the line and identify two points. Let's take Points A and B as the two points with coordinates (0,6) and (2,0) respectively, which lie on the line.

Step 3: Compute the Slope

The slope of the line is computed using the formula \(m = (y2 - y1) / (x2 - x1)\). So, substituting the coordinates of points A and B into this formula, we get \(m = (0 - 6) / (2 - 0) = -3\).

Step 4: Check the Result

The slope calculated from the points on the graph is -3. This should be compared with the coefficient of \(x\) in the equation, which is also -3. Thus, the calculations have been verified.

Key Concepts

Graphing Equations

The first step in understanding slopes and lines is graphing their equations. Graphing equations like \( y = -3x + 6 \) involves plotting points on a coordinate plane. The graphing utility makes this task easier by allowing you to directly input the equation and instantly see the line that represents it.

When you graph this equation, you're essentially translating a mathematical expression into a visual line that reflects its properties, such as slope and intercepts. The line \( y = -3x + 6 \) will cross the y-axis at 6, because when \( x = 0 \), \( y = 6 \).

This initial visualization helps to set the stage for further exploring the line's characteristics, like finding specific points that lie on the line, which is crucial for identifying the slope through practical examples.

Coordinate Geometry

Coordinate geometry allows us to explore the properties and relationships of geometric figures using a coordinate plane. It's the essential framework for understanding the slope of a line, which measures the steepness or direction of the line.

To find two points on the line, we use the coordinate plane and its axes to express these points in terms of \( (x, y) \) coordinates. By using the TRACE feature of a graphing utility, you can freely move along the line to pick coordinates like (0,6) and (2,0) for points A and B.

• (0,6) is where the line intersects the y-axis.
• (2,0) is where the line intersects the x-axis.

Coordinate geometry then allows us to use these points to calculate the slope \( m \) of the line with the formula \( m = (y_2 - y_1) / (x_2 - x_1) \). This expression determines how much \( y \) changes for every unit \( x \) changes, clarifying the line's slant.

Using Graphing Utilities

Graphing utilities are powerful tools that simplify the process of visualizing and analyzing mathematical equations. With a graphing utility, you can quickly turn equations into graphs, enhancing your understanding by directly observing the line's behavior.

A primary feature of graphing utilities is the ability to trace along a plotted line. By using the TRACE function, you can select and explore specific points on the line. This feature is particularly helpful in identifying points for slope calculation, as it automates some of the more tedious aspects of manual plotting.

• Enter the line's equation into the utility.
• Utilize the TRACE feature to find useful points.

Moreover, graphing utilities allow you to instantly compare your calculated results with analytical solutions. In our example, after calculating the slope using the points from the TRACE function, you can compare it to the coefficient of \( x \) in the equation to verify accuracy.